Hilbert Series and Operator Basis for NRQED and NRQCD/HQET

We use a Hilbert series to construct an operator basis in the $1/m$ expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET). We present a list of effective operators with mass dimension $d\leq 8$. Comparing to the current literature, our results for NRQED agree for $d\leq 8$, but there are some discrepancies in NRQCD/HQET at $d=7$ and 8.

There is currently no disagreement in the literature regarding the number of NRQED operators up to and including order 1/m 4 , and the Hilbert series we construct for NRQED agrees with the results in Refs. [8][9][10][11]. Also, our results for NRQCD/HQET agree with those in Refs. [8,13] up to order 1/m 3 , but we find discrepancies with other analyses at 1/m 3 and 1/m 4 . Specifically, we count 11 operators at 1/m 3 (as does Ref. [8]), and 25 operators at 1/m 4 . However, at order 1/m 3 , Ref. [18] says there are 5, and Refs. [14,15] claim there are 9. At order 1/m 4 , Refs. [14,15] claim there are 18 operators. The differences between our results and those found in Refs. [14,15] could be explained by there being two symmetric SU (3) color singlets for operators with two gauge bosons. We discuss this further in Section V.

II. EFFECTIVE THEORY FOR A NONRELATIVISTIC FERMION
We consider a system where the relevant dynamics of a massive fermion in an external, dynamical, gauge field occurs at energy scales well below the rest mass, m, of a fermion. 2 The following effective Lagrangian can be used to describe such a system with a heavy fermion: (1) Here, ψ is a two-component Pauli spinor, c k is a coupling constant, and O k are Hermitian operators, suppressed by the appropriate powers of 1/m. All operators O k must be rotationally and translationally invariant, contain either zero or one spin vector s i , and are built from time and spatial components of covariant derivatives, i.e., iD t and iD ⊥ , respectively.
Listing all operators that satisfy only these conditions leads to over counting, since some operators can be related to others via integration by parts or relations associated with the equations of motion. In particular, operators with derivatives that act on ψ † can be related to other operators with derivatives that act on ψ by integrating by parts: where O is some operator. Also, the equation of motion for ψ is Therefore, if D t acts on ψ, it can be replaced by a series of operators, all at higher powers in 1/m: where O is some Hermitian operator. A similar argument holds for the equations of motion associated with ψ † . There are also equations of motion associated with the external gauge field. We refer to antisymmetric combinations of covariant derivatives as The equations of motion for E i and B i are Maxwell's equations, or its non abelian version: where ρ and J are the external charge and current densities, respectively. In summary, correct enumeration of operators, when accounting for redundancies associated with integration by parts and the equations of motion, amounts to removing: (1) total derivatives, (2) those of the form More symmetry is expected in a theory with a nonrelativistic fermion, such as reparameterization invariance [8,19] or residual Lorentz symmetry [20]. Imposing this invariance would require establishing relationships between the coefficients of operators at different orders in 1/m. For this work, however, we focus only on a rotationally-and translationally-invariant theory, since this can be readily encoded into a Hilbert series. In the particular examples of NRQED and NRQCD/HQET, invariance under parity and time reversal transformations are also expected, since the underlying theories are invariant under parity and time reversal, which we discuss in Sections IV and V.

III. HILBERT SERIES FOR A NONRELATIVISTIC THEORY
The Hilbert series can be used to count the number of invariants under a group transformation, utilizing the plethystic exponential, defined as (−1) n+1 ψ n n χ R (z n 1 , z n 2 , ..., z n k ) .
Here, χ R is the character of the representation R of group G of rank k, φ and ψ are spurions (complex numbers taken to have modulus less than unity) corresponding to the field associated with the representation R, and the z i 's are complex numbers with unit modulus (called fugacities) that parameterize the maximal torus of G. The plethystic exponentials are defined so as to ensure, if Taylor expanded in φ or ψ, that the nth power of φ or ψ will have a coefficient equal to the character of symmetric (in the case of bosonic statistics) or antisymmetric (in the case of fermionic statistics) tensor products, constructed out of representation R, n times. The Hilbert series that counts the total number of group invariants is generated by performing the following integral (often called the Molien-Weyl formula): where the contour integral is done over the maximal torus of the group G with respect to the Haar measure, [dµ] G , associated with the group G. The Hilbert series, as defined by Eq. (12), is a polynomial in the spurions such that the coefficient of different powers of the spurions counts the number of invariants under the group G. 3 For further details, we refer to [1][2][3][4][5].
Using the machinery of the Hilbert series, we can construct all possible operators O k in Eq.
(1). The characters for E, B, ψ, ψ † , and s are (note that P 0 , P ⊥ , D t , and D ⊥ are defined in Eqs. (21) and (22) and in the text thereafter): where χ are the characters for a 3 of SO(3), a 2 of SU (2), and a 3 of SU (2), 3 The invariants are counted using the character orthogonality relation: where χR and χ R are characters of irreducible representations, R and R , of G. When R is a trivial singlet representation, using χ singlet = 1, we have Therefore, using the definition of the plethystic exponentials, the Hilbert series we use, which counts the invariants under group G is respectively. Explicit expressions for these characters can be found in Appendix A. The characters χ C represent the way E, B, ψ, and ψ † are charged under the external gauge field. For example, if the fermion has color, then χ C E and χ C B are both the characters for the adjoint representation of SU (3), and χ C ψ (χ C ψ † ) is the character for the fundamental (antifundamental) representation of SU (3). P 0 and P ⊥ generate all symmetric products of temporal and spatial derivatives, respectively: where D t and D ⊥ are the spurions that correspond to time and spatial derivatives in the operator, respectively. The characters in Eqs. (16) (16), to enforce the constraint that ∇ · (∇ × E) = 0. Without it, Hilbert series will erroneously subtract off ∇ · (∇ × E), which was not there to begin with before the subtraction.
The general Hilbert series for a theory with a heavy fermion is The bosonic plethystic exponential, i.e., Eq. (10), is used for E, B, and s, while the fermionic one, i.e., Eq. (11), is used for ψ and ψ † . The expressions for the Haar measures can be found in Appendix A. The factor of 1/P 0 P ⊥ removes operators that are total time derivatives and total spatial derivatives. 4 This method, however, will over-subtract operators that are total derivatives, but which have already been subtracted by the equations of motion. Thus, this Hilbert series will, in general, produce some terms with negative signs, all of which are redundant operators, and can be ignored. One can expand the plethystic exponentials for ψ and ψ † to first order, and perform the SU (2) integral by hand, which results in the Hilbert series for the operators O k in Eq. (1): Explicit expressions for the Hilbert series NRQED and NRQCD/HQET will be given in Sections IV and V, respectively, including discussions on how to impose invariance under parity and time reversal.

IV. NRQED
In NRQED, the relevant gauge symmetry group is U (1). Here, χ C E = χ C B = 1, since photons do not have any U (1) charge, and The Hilbert series for O k in Eq. (1) in NRQED is Again, we ignore any negative terms generated by this Hilbert series, since they are both total derivatives and related to other operators by the equations of motion, as discussed in Section III. Since parity is a symmetry of QED, one can demand that O k respects parity by requiring that it is composed of any number of parity-even objects, i.e., D t , B, and s, and an even number of parity-odd objects, i.e., D ⊥ and E. This can be automated without explicitly constructing the operators O k by hand.
The output for this Hilbert series for dimensions 5, 6, 7, and 8, before imposing invariance under time reversal, is  Table II for d = 5, 6, 7, and 8. T -even operators are those with any number of T -even objects, i.e., E and ∂ ⊥ , and an even numbers of T -odd objects, i.e., ∂ t , iD ⊥ , B, and s.
In the special case of NRQED, where the group is abelian, there is a method to impose T invariance that is easily automated. This is done by modifying the Hilbert series to distinguish those spatial derivatives ∂ ⊥ acting only on E and B from the spatial derivatives iD ⊥ that act on ψ. Here, the former ones are always T -even, while the latter are always T -odd. This results in: This method agrees with the result when explicitly constructing operators and selecting by hand only those that are T -even, and it agrees with the lists of operators up to and including d = 8 in Refs. [8][9][10][11].
It is straight forward, using the Hilbert series as a guide, to explicitly list operators for d > 8. We show in Fig. 1 the total number of operators in NRQED up to d = 18, when T invariance is imposed, and list the total number of operators in Table I.    (3) indices. Note that these Hermitian operators O are those in the bilinear ψ † Oψ, and the square brackets indicate that the derivative acts only on the object in the square bracket. Also, in the special case of NRQED, time-reversal symmetry can be imposed in an automated way, without constructing Hermitian operators by hand; see the text at the end of Section IV for details.

V. NRQCD/HQET
The construction of the Hilbert series for NRQCD/HQET is very similar to that of NRQED, where . The Hilbert series for the operators O k in Eq.
(1) is When invariance under parity is imposed, the output from this Hilbert series for operators of mass dimension d = 5, 6, 7, and 8 is: Unlike NRQED, we have not found an automated way to implement invariance under time reversal in NRQCD/HQET, because T acts as an anti-unitary operator, and counting T -invariant operators requires keeping track of factors i while constructing Hermitian operators. This is not an issue when constructing invariants in NRQED, since it has an abelian U (1) symmetry, but when the group is non-abelian, like SU (3), the algebra's structure constants, e.g., f abc , bring with them a factor of i, and imposing Tsymmetry is no longer straight-forward.
We take the output from this Hilbert series and explicitly contract indices by hand, separating those that are even and odd under time reversal. The prescription is very close to the one we used in NRQED.
We choose to suppress color indices, and express E = E a T a and B = B a T a , where T a are the eight generators of SU (3), which satisfy: We utilize the following notation, where the letters i, j, k, l, m, ... are used for SO(3) indices, and the letters a, b, c, ... are used to signify the SU (3) generators: where A µ ≡ A µ a T a is the gauge field. When there are two SU (3) generators in an operator, one can use the relation that results in adding Eqs. (40) and (41) together: From this, one can see that, for example, the operator E 2 in the Hilbert series can be contracted in two ways: A third contraction with f abc is completely antisymmetric in a, b, c, which results in an operator equal to zero, in this case. Finally, it should be noted that f abc should be thought of a odd under time reversal. 5 The two contractions in Eq. (44) give rise to different matrix elements, since the contraction of color indices would be different. A complete list of NRQCD/HQET operators can be found in Table III for d ≤ 8. Extending the list to higher orders would be straight-forward.
Our results agree with those in Ref. [8] for NRQCD/HQET operators up to and including operators of order 1/m 3 . However, we find a different number compared to Refs. [14,15] for operators at order 1/m 3 and 1/m 4 . Specifically, Refs. [14,15] claim there are 9 operators at 1/m 3 , and 18 operators at 1/m 4 , while we find 11 and 25, respectively. These discrepancies are consistent with the possibility that Refs. [14,15] count only once the two symmetric terms, i.e., contractions with δ ab and d abc , in Eq. (43).

VI. DISCUSSION AND CONCLUSIONS
We construct a Hilbert series for an effective theory with a single non-relativistic fermion in an external, and dynamical, gauge field, defining characters and using a method to subtract operators that are related to others via the equations of motion associated with the heavy fermion and the external gauge bosons,  Table II for NRQED and Table III for NRQCD/HQET. In a theory with a nonrelativistic fermion, additional symmetry, e.g., reparameterization invariance [8,19] or residual Lorentz symmetry [20], is expected, in general. However, we do not impose such additional constraints, since it remains an open question regarding how to encode such requirements with Hilbert-series methods.
Our results agree with those presented in Refs. [8][9][10][11] for NRQED, which discuss operators up to and including d = 8. The total number of operators in NRQED grows exponentially, as shown Fig. 1 and listed in Table I for mass dimension d ≤ 18. When using a Hilbert series for NRQCD/HQET, we count a total of 2 operators each at orders 1/m and 1/m 2 , which agrees with Ref. [13], 11 operators at 1/m 3 , which agrees with Ref. [8], and 25 operators at 1/m 4 . However, at order 1/m 3 , other analyses claim   III: Same as Table II, but for NRQCD/HQET, separating those operators that are even and odd under time reversal. See the text at the end of Section V for a discussion regarding notation.
that there are either 5 [18], or 9 [14,15] total operators, and Refs. [14,15] claim there are a total of 18 operators at 1/m 4 . The differences between our results and those found in Refs. [14,15] can be explained by the existence of two symmetric SU (3) color singlets for operators with two gauge bosons, as discussed at the end of Section V. It is possible that analyses using the results in Refs. [14,15,18], may need to be reevaluated, e.g., Refs. [16,17,21].
The authors of Refs. [1, 2] discuss a connection between enumerating operators in a relativistic effective theory and the representations of the relativistic conformal group. Here, selecting only primary operators constructed out of tensor products of the conformal group's short representations correctly accounts for redundancies between operators via integration by parts and the equations of motion. We strongly suspect that our results can be reformulated in terms of the non-relativistic conformal group [22][23][24][25][26], and we take this up as future work.
Note: While this article was in review for publication, the authors of Ref. [15] updated their work, and their results now agree with our enumeration of NRQCD/HQET effective operators for d ≤ 8.